Pick the relation: C = C₀·e−kt, LD = Vd·C/(S·F), maintenance rate = CL·Css/(S·F), τ = ln(Cpeak/Ctrough)/k.
For kidney function, estimate CrCl with Cockcroft–Gault first, then scale the dose or the interval.
Check the result: levels fall by half every half-life, and steady state takes about five half-lives.
Pharmacy calculations
Percentage strength
% w/v = g per 100 mL
Amount in a solution
g = % × mL / 100
Ratio strength
1:X = 1 g in X mL; % = 100 / X
Reconstitution
final volume = diluent + powder volume
Concentration
mg/mL = drug in vial / final volume
Alligation
parts of H = T − L, parts of L = H − T
Water or base
0%
Pediatric & critical-care dosing
Daily dose
mg/day = mg/kg/day × weight (kg)
Doses per day
number = 24 h / dosing interval (h)
Pounds to kilograms
kg = lb / 2.2
Infusion concentration
units/mL = units in bag / bag volume
Pump rate
mL/h = dose per hour / concentration
Titrated drip
mL/h = mcg/kg/min × kg × 60 / (mcg/mL)
Dose from pump rate
mcg/kg/min = mL/h × (mcg/mL) / (60 × kg)
Units
1 mg = 1000 mcg
2Practice problems
1.Clinical pharmacokineticsMedium
Practice exercise (made-up values, not clinical guidance). A patient receives an IV bolus of 190 mg of gentamicin. The level back-extrapolated to time zero is 9 mg/L and the half-life is 3.5 h. Find the volume of distribution Vd.
D = 190 mgC₀ = 9 mg/Lt½ = 3.5 h
Vd = L
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A theophylline infusion is started without a loading dose. The half-life in this patient is 8 h. How long t does it take to reach 75% of the steady-state level?
t½ = 8 h75% Css
t = h
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:1000. You need 0.69 mg. Draw up the volume in the 1 mL syringe: what volume V is that?
1:10000.69 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 250 mg of powder. You add 1.9 mL of diluent, and the powder adds 0.6 mL of volume. The prescription is 120 mg. Draw up the dose: what volume V do you give?
250 mg1.9 mL + 0.6 mL120 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A bag contains 12 500 units of heparin in 250 mL. The order is 16 units/kg/h for a patient who weighs 68 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
12 500 units/250 mL16 units/kg/h68 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A bag contains dobutamine 500 mg in 250 mL. The order is 2.5 mcg/kg/min for a patient who weighs 76 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
500 mg/250 mL2.5 mcg/kg/min76 kg
R = mL/h
7.PharmacologyMedium
During a rapid infusion of vancomycin, a patient develops flushing and a red rash on the face, neck and upper body. What is the most likely cause and action?
AAn infusion reaction; stop or slow the infusion and inform the prescriber
BAcute kidney failure; stop all IV fluids and antibiotics at once
CA sign that the infection is spreading; give a second dose
DA normal effect; increase the infusion rate to finish sooner
8.PharmacologyHard
Which heart failure drug improves symptoms and reduces hospital admissions but has not been shown to reduce mortality?
AA mineralocorticoid receptor antagonist such as spironolactone
BA beta blocker such as bisoprolol
CAn ACE inhibitor
DDigoxin
3Solutions
1.Clinical pharmacokinetics
Answer: Vd = 21.11 L
Clearance and volume: at steady state the dosing rate equals the elimination rate, so the maintenance rate = CL × Css. A loading dose is LD = Vd × C/(S × F), with S the salt factor (fraction of active drug in the salt) and F the bioavailability (1 for IV). From two levels, k = ln(C₁/C₂)/Δt; for an IV bolus Vd = dose/C₀ and CL = k × Vd. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Volume of distribution
Vd = 190/9 = 21.11 L
Answer
Vd = 21.11 L
2.Clinical pharmacokinetics
Answer: t = 16 h
Designing a regimen: with rapid IV doses the level falls from peak to trough during the interval, so τ = ln(Cpeak/Ctrough)/k and the dose that restores the peak is Vd × (Cpeak − Ctrough). The average steady-state level is Css,avg = S·F·dose/(CL·τ). An infusion rises as C = (R/CL)(1 − e−kt). For a drug cleared by the kidneys, the dose is scaled with the creatinine clearance. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Rate constant
k = 0.693/8 = 0.08664 h⁻¹
Time to the given fraction of steady state
t = −ln(1 − 0.75)/0.08664 = 1.386/0.08664 = 16 h
Answer
t = 16 h
That is 2 half-lives.
3.Pharmacy calculations
Answer: V = 0.69 mL
Strength to volume: turn the strength into mg per mL first. A w/v percentage is grams per 100 mL, so x% = 10x mg/mL. A ratio strength 1:n is 1 g in n mL. After reconstitution the concentration is the powder's dose over the final volume (diluent + powder volume). Then volume = dose/concentration.
Strength in mg/mL
1000 mg/1000 mL = 1 mg/mL
Volume
V = 0.69/1 = 0.69 mL
Answer
V = 0.69 mL
4.Pharmacy calculations
Answer: V = 1.2 mL
Strength to volume: turn the strength into mg per mL first. A w/v percentage is grams per 100 mL, so x% = 10x mg/mL. A ratio strength 1:n is 1 g in n mL. After reconstitution the concentration is the powder's dose over the final volume (diluent + powder volume). Then volume = dose/concentration.
Final volume and concentration
1.9 + 0.6 = 2.5 mL
C = 250/2.5 = 100 mg/mL
Volume
V = 120/100 = 1.2 mL
Answer
V = 1.2 mL
5.Pediatric & critical-care dosing
Answer: R = 21.8 mL/h
Heparin and insulin infusions: concentration = units in the bag/bag volume (units/mL). Pump rate (mL/h) = units per hour/concentration, and units per hour = mL/h × concentration. For a weight-based order, units/h = units/kg/h × weight first. A bolus volume = units needed/vial strength. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Units per hour
16·68 = 1 088 units/h
Concentration
12 500/250 = 50 units/mL
Pump rate
R = 1088/50 = 21.76 ≈ 21.8 mL/h
Answer
R = 21.8 mL/h
6.Pediatric & critical-care dosing
Answer: R = 5.7 mL/h
Titrated drips: first find the concentration in mcg/mL (mg × 1000/mL). For a weight-based order, mcg/min = mcg/kg/min × weight. Then mL/h = mcg/min × 60/concentration. Working backwards, mcg/kg/min = mL/h × concentration/(60 × weight). The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Concentration
500 mg = 500 000 mcg
500 000/250 = 2000 mcg/mL
Dose per minute and per hour
2.5·76 = 190 mcg/min
190·60 = 11400 mcg/h
Pump rate
R = 11400/2000 = 5.7 mL/h
Answer
R = 5.7 mL/h
7.Pharmacology
Answer: A · An infusion reaction; stop or slow the infusion and inform the prescriber
Why this is the answer
Rapid vancomycin infusion releases histamine, causing flushing and itching of the upper body (formerly called "red man syndrome"). It is managed by stopping or slowing the infusion, and sometimes with an antihistamine.
Answer
A · An infusion reaction; stop or slow the infusion and inform the prescriber
8.Pharmacology
Answer: D · Digoxin
Why this is the answer
ACE inhibitors (or ARBs/ARNI), beta blockers, mineralocorticoid antagonists and SGLT2 inhibitors reduce mortality in heart failure with reduced ejection fraction. Digoxin improves symptoms and reduces admissions without reducing mortality.